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A polynomial-based QCQP solver for encrypted optimization

Sebastian Schlor, Andrea Iannelli, Junsoo Kim, Hyungbo Shim, Frank Allgöwer

TL;DR

This work addresses privacy-preserving optimization of convex QCQPs under homomorphic encryption by replacing hard constraints with a sequence of polynomial penalties and solving a sequence of unconstrained problems via gradient descent. The authors prove that each auxiliary minimizer stays inside the feasible set and that the sequence of minimizers converges to the original constrained minimizer as the penalty degree grows, while ensuring positive invariance of the feasible set under iterations. The method is tailored to encrypted computation, requiring only additions and multiplications, and includes an encrypted implementation discussion alongside a concrete min(a,b) example to illustrate accuracy versus iteration count. The approach offers a practical pathway to perform constrained optimization on private data without trusted third parties, with potential impact on privacy-preserving control and MPC applications using homomorphic encryption.

Abstract

In this paper, we present a novel method for solving a class of quadratically constrained quadratic optimization problems using only additions and multiplications. This approach enables solving constrained optimization problems on private data since the operations involved are compatible with the capabilities of homomorphic encryption schemes. To solve the constrained optimization problem, a sequence of polynomial penalty functions of increasing degree is introduced, which are sufficiently steep at the boundary of the feasible set. Adding the penalty function to the original cost function creates a sequence of unconstrained optimization problems whose minimizer always lies in the admissible set and converges to the minimizer of the constrained problem. A gradient descent method is used to generate a sequence of iterates associated with these problems. For the algorithm, it is shown that the iterate converges to a minimizer of the original problem, and the feasible set is positively invariant under the iteration. Finally, the method is demonstrated on an illustrative cryptographic problem, finding the smaller value of two numbers, and the encrypted implementability is discussed.

A polynomial-based QCQP solver for encrypted optimization

TL;DR

This work addresses privacy-preserving optimization of convex QCQPs under homomorphic encryption by replacing hard constraints with a sequence of polynomial penalties and solving a sequence of unconstrained problems via gradient descent. The authors prove that each auxiliary minimizer stays inside the feasible set and that the sequence of minimizers converges to the original constrained minimizer as the penalty degree grows, while ensuring positive invariance of the feasible set under iterations. The method is tailored to encrypted computation, requiring only additions and multiplications, and includes an encrypted implementation discussion alongside a concrete min(a,b) example to illustrate accuracy versus iteration count. The approach offers a practical pathway to perform constrained optimization on private data without trusted third parties, with potential impact on privacy-preserving control and MPC applications using homomorphic encryption.

Abstract

In this paper, we present a novel method for solving a class of quadratically constrained quadratic optimization problems using only additions and multiplications. This approach enables solving constrained optimization problems on private data since the operations involved are compatible with the capabilities of homomorphic encryption schemes. To solve the constrained optimization problem, a sequence of polynomial penalty functions of increasing degree is introduced, which are sufficiently steep at the boundary of the feasible set. Adding the penalty function to the original cost function creates a sequence of unconstrained optimization problems whose minimizer always lies in the admissible set and converges to the minimizer of the constrained problem. A gradient descent method is used to generate a sequence of iterates associated with these problems. For the algorithm, it is shown that the iterate converges to a minimizer of the original problem, and the feasible set is positively invariant under the iteration. Finally, the method is demonstrated on an illustrative cryptographic problem, finding the smaller value of two numbers, and the encrypted implementability is discussed.
Paper Structure (19 sections, 8 theorems, 39 equations, 1 figure)

This paper contains 19 sections, 8 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Contribution
  4. Notation
  5. Problem setup and main idea
  6. Problem setup
  7. Main idea
  8. Sequence of unconstrained problems
  9. Design of the unconstrained problems
  10. Analysis of the unconstrained problems
  11. Sequential gradient descent
  12. Encrypted implementation
  13. Example: $\min(a,b)$
  14. Accuracy of the auxiliary solution
  15. Accuracy of a single gradient step
  16. ...and 4 more sections

Key Result

Lemma 1

For every $k\in\mathbb{N}$, Problem eq:AuxiliaryProblem has a unique solution.

Figures (1)

  • Figure 1: Example for the sequence of cost functions $\{J_k\}_k$ based on the original cost function $f$ and the sequence of penalty functions $\{p_k\}_k$ for the set $\mathcal{C}$ for $k\in\{1,2,3, N\}$ with $N=15$. The sequence of minimizers $\{x_k^\star\}_k$ converges to the minimizer $x^\star$ of the constrained problem as $k\to\infty$ (cf. Proposition \ref{['lem:auxMin:convegence']}).

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Lemma 3
  • proof
  • Remark 1
  • Proposition 2
  • ...and 8 more